The time horizon for stochastic homogenization of the one-dimensional wave equation

2.1. Homogenized coefficients and correctors

In the one-dimensional case, the homogenized coefficients are given by simple formulas. Essentially, the effective density ρ ¯ is the arithmetic mean of ρ and the effective permeability a ¯ is the harmonic mean of a. Since we also consider non-ergodic media, we have a very general dependence on the spatial variable x ∈ R n ; this requires slightly more involved definitions. In any case, the stochastic setting allows to take expectations: In the probability space ( Ω P , A , P ) we denote the expected value of a random variable f : Ω P → R with brackets ⟨ . ⟩ and set ⟨ f ⟩ := ∫ Ω P f ( ω ) d P ( ω ) .

Since we do not impose stationarity or ergodicity of the medium, our definition of effective coefficients involves several averages. In one space dimension, we use the following concept.

Definition 2.1 (Effective coefficients in the sense of averages)

For ρ , a : R × Ω P → R , we define the effective coefficients in the sense of averages as

ρ ¯ := lim | r | → ∞ ⟨ − ∫ 0 r − ∫ 0 y ρ ( s ) d s d y ⟩ , a ¯ := lim | r | → ∞ ⟨ − ∫ 0 r − ∫ 0 y 1 / a ( s ) d s d y ⟩ − 1 ,

(2.1)

demanding on the model functions ρ and a that the two limits of (2.1) exist. Note that we use a special sign convention for integrals, see (2.2) below.

In (2.1) and similar expressions in the subsequent text, we are interested in averages of the integrand. This means that, when we write ∫ 0 y g ( s ) d s for negative y, we are indeed interested in the value of ∫ [ y , 0 ] g ( s ) d s . In other words, we want that integrals of positive functions are positive, even when the limits of the integrand are not ordered in the usual way. We therefore set, for y < 0 ,

∫ 0 y g ( s ) d s := ∫ y 0 g ( s ) d s and − ∫ 0 y g ( s ) d s := | y | − 1 ∫ y 0 g ( s ) d s .

(2.2)

For periodic and, more generally, for stationary media, (2.1) can be replaced by simpler expressions. For stationary media, expectations are independent of y, hence ρ ¯ = ⟨ ρ ( y , . ) ⟩ and a ¯ = ⟨ 1 / a ( y , . ) ⟩ − 1 for arbitrary y ∈ R . When the stochastic medium is ergodic, then spatial averages converge to expected values, hence, in this case, it is not necessary to take expectations in (2.1). For periodic media, the formula is even simpler: One can omit the expectation and integrate only over one periodicity cell.

In the one-dimensional setting, we can write the two wave equations of interest as follows. For coefficients ρ, a given by (1.2) and rescaled coefficients ρ ε , a ε given by (1.3) and a source f : R × R + → R , we consider the sequence of solutions u ε : R × R + → R of the ε -problem

◻ ε u ε := ρ ε ∂ t 2 u ε − ∂ x ( a ε ∂ x u ε ) = f with u ε ( ⋅ , 0 ) = ∂ t u ε ( ⋅ , 0 ) = 0 ,

(2.3)

◻ ¯ u ¯ := ρ ¯ ∂ t 2 u ¯ − ∂ x ( a ¯ ∂ x u ¯ ) = f with u ¯ ( ⋅ , 0 ) = ∂ t u ¯ ( ⋅ , 0 ) = 0.

(2.4)

We turn to the construction of correctors and harmonic coordinates. The one-dimensional corrector equation reads ∂ y [ a ( y ) ( 1 + ∂ y Φ ( y ) ) ] = 0 for y ∈ R . For a given coefficient field a, one seeks a solution Φ : R → R of this equation. In one space dimension, the solution is given by an integral: Since the argument in the squared brackets must be independent of y, the squared bracket coincides with some constant. The constant value is indeed the effective coefficient, a ( y ) ( 1 + ∂ y Φ ( y ) ) = a ¯ for every y ∈ R . Dividing by a ( y ) and integrating over y, this formula can be used to define Φ. The procedure for the corrector Ψ for ρ is similar.

Definition 2.2 (Correctors)

Let ρ , a : R × Ω P → ( 0 , ∞ ) be stochastic coefficients and let ρ ¯ and a ¯ be two positive numbers. We define corresponding correctors as

Ψ ( y ) := Ψ ( y , ω ) := ∫ 0 y { ρ ( s ) ρ ¯ − 1 } d s , Φ ( y ) := Φ ( y , ω ) := ∫ 0 y { a ¯ a ( s ) − 1 } d s ,

(2.5)

where we suppressed the argument ω also in ρ and a.

Dependence on coefficients. We note that the correctors depend on the coefficients ρ and a and on the effective coefficients ρ ¯ and a ¯ . Indeed, one can regard ρ ¯ and a ¯ as being chosen in such a way that the correctors have minimal growth.

Rescaling. Rescaled coefficients are defined by ρ ε ( x ) = ρ ( x / ε ) and a ε ( x ) = a ( x / ε ) . The correctors are rescaled as Φ ε ( x ) := ε Φ ( x / ε ) and Ψ ε ( x ) := ε Ψ ( x / ε ) . We note that Φ ε satisfies, for every x ∈ R ,

∂ x [ ( a ε ( x ) ( 1 + ∂ x Φ ε ( x ) ) ] = 0 ,

a ε ( x ) ( 1 + ∂ x Φ ε ( x ) ) = a ¯ .

(2.6)

In this article, we study the question whether or not the homogenized solution u ¯ is a good approximation for the heterogeneous media solution u ε . The answer depends on the time span of interest and on the “quality” of the stochastic medium. We will measure the latter in terms of growth rates of the correctors.

Let ( Ω P , A , P ) be a probability space, let ρ and a be stochastic coefficients, and let the space dimension be d = 1 .

Definition 2.3 (Model class parameter γ)

For fixed Λ ⩾ 1 we consider maps ρ , a : R × Ω P → [ 1 Λ , Λ ] . Given ρ and a and two positive numbers ρ ¯ and a ¯ , we denote the corresponding correctors as Ψ and Φ, see Definition 2.2. We define the set of supercritical parameters γ ′ as

Γ s c := { ∫ 0 r γ ′ ∈ [ 0 , ∞ ] | ∃ C , ρ ¯ , a ¯ ∈ R ∀ r ∈ R : ⟨ − ∫ 0 r | Ψ ( y ) | 2 d y ⟩ + ⟨ − ∫ 0 r | Φ ( y ) | 2 d y ⟩ ⩽ C ( 1 + | r | γ ′ ) 2 ∫ 0 r } .

(2.7)

The model class parameter γ is set to be

γ := inf Γ s c .

(2.8)

With this definition, we associate to a model (given by random fields a and ρ) a model class parameter γ = γ ( ρ , a ) .

Remarks. 1.) The class is a number γ ∈ [ 0 , 1 ] . We recall that we always assume boundedness of a and ρ. Under this assumption, the functions Φ and Ψ have at most linear growth, which implies ( 1 , ∞ ) ⊂ Γ s c and hence γ ⩽ 1 . By definition, there always holds γ ⩾ 0 .

2.) Interpretation and periodic media. With γ ∈ [ 0 , 1 ] defined as above, we say that ( ρ , a ) defines a model of class γ. The quantity γ quantifies the sublinearity of the correctors. We recall the periodic coefficients have bounded correctors, hence the model parameter for periodic coefficients is γ = 0 .

3.) Effective coefficients in the sense of optimal correctors. The definition can also be used to define effective coefficients: The numbers ρ ¯ and a ¯ for which the optimal growth rate in (2.7) is obtained are the effective coefficients in the sense of optimal correctors.

We emphasize that the two possible definitions of effective coefficients are closely related. We will show in Lemma 4.4 the following: When the growth of (2.7) holds for ρ ¯ and a ¯ and for a number γ ′ < 1 , then ρ ¯ and a ¯ are the effective coefficients in the sense of averages of Definition 2.1.

Example 2.4 (The simplest stochastic medium)

Let ( a j ) j ∈ Z and ( ρ j ) j ∈ Z be i.i.d. random variables, chosen with a uniform distribution in [ 1 , 2 ] . We define a and ρ by setting a ( x ) = a j and ρ ( x ) = ρ j for x ∈ [ j , j + 1 ) , and we set ρ ¯ = ⟨ ρ 0 ⟩ and a ¯ := ⟨ a 0 − 1 ⟩ − 1 . This defines a stochastic model. The model class parameter of this model is γ = 1 / 2 .

Let us sketch how to calculate the model parameter γ. For large y ∈ N , the quantities Φ ( y ) and Ψ ( y ) of (2.5) are sums of y i.i.d. random variables with vanishing expected value, in particular we have for Φ ( y ) , with y ∈ R ,

| y | σ 2 − 1 ⩽ ⟨ | Φ ( y ) | 2 ⟩ ⩽ | y | σ 2 + 1 , σ 2 := ⟨ ( a ¯ a 0 − 1 ) ⟩ 2

(for Ψ ( y ) holds an analogous estimate with σ 2 := ⟨ ( ρ 0 ρ ¯ − 1 ) 2 ⟩ ). We conclude that the expressions in (2.7) behave like (we write the formulas for Φ and y > 0 ):

σ 2 2 y − 1 = − ∫ 0 y σ 2 s − 1 d s ⩽ ⟨ − ∫ 0 y | Φ ( s ) | 2 d s ⟩ ⩽ − ∫ 0 y σ 2 s + 1 d s = σ 2 2 y + 1.

The above estimate, together with the analogous estimate for Ψ , yield γ = 1 / 2 .

For media with growth parameter γ ∈ ( 1 / 2 , 1 ) , we refer to Appendix C.

We have to make precise what we mean with the sloppy expression “homogenization works”. The subsequent gives a definition. On the one hand, we have to introduce the parameter β that measures the time span under consideration. On the other hand, we have to choose a notion of convergence. Regarding the latter, it seems natural to work with the norms that are suggested by energy estimates, i.e., we use an L ∞ -norm in time, and measure first derivatives with the norm L 2 in the spatial coordinate. We actually leave out first spatial derivatives, since one expects oscillations of ∂ x u ε , but no oscillations of ∂ x u ¯ . To compare only time derivatives turns out to be the more robust concept.

Definition 2.5 (Homogenization time parameter β)

Let ρ , a : R × Ω P → R be stochastic coefficients and let β ∈ [ 0 , ∞ ) be a positive parameter. We say that classical homogenization works with parameter β if the following holds: For any f ∈ C 2 ( R × R + , R ) with compact support and any T 0 > 0 , the solutions u ε of (2.3) and u ¯ of (2.4) satisfy

lim ε → 0 sup t ∈ [ 0 , T 0 ε − β ] ⟨ ‖ ∂ t u ε ( ⋅ , t ) − ∂ t u ¯ ( ⋅ , t ) ‖ L 2 ( R ) ⟩ = 0.

(2.9)

Loosely speaking: β ⩾ 0 is the parameter such that classical stochastic homogenization works on time intervals [ 0 , T 0 ε − β ] .

We can now formulate our first main result.

Theorem 2.6 Critical parameters

Let γ ∈ [ 0 , 1 ] be a number such that a model of that class exists. With the critical parameters

β − := 1 − γ 1 + γ , β + := 1 − γ γ

(2.10)

there holds: (1)

For β ∈ [ 0 , β − ) : For all coefficients ( ρ , a ) of class γ, classical homogenization works with parameter β.

(2)

For β ∈ ( β + , ∞ ) : There exist coefficients ( ρ , a ) of class γ such that classical homogenization does not work with parameter β.

The main novelty of Theorem 2.6 is Part (2), while Part (1) follows by well established computations.

Section 3 is devoted to the first claim of the theorem, the positive homogenization result. Section 4 is devoted to the second claim of the theorem, the negative result.

2.3. The case of matched impedance

The main novelty of Theorem 2.6 is Part (2), which concerns the failure of classical homogenization on sufficiently large time spans. In the special class of media with matched impedance one can understand well the mechanism of this failure. It is related to a large likelihood of a wrong averaged wave speed on a considerable time span. We would like to mention that we learned the techniques for matched impedance media from [14]. Below we provide an informal (but detailed) discussion of the failure of homogenization in media with matched impedance; rigorous results in this setting can be found in Section 5.

A medium has matched impedance if the product ρ ⋅ a is a constant function. Here, we assume that ρ ( x , ω ) ⋅ a ( x , ω ) = 1 holds for all x ∈ R and for P -almost every ω ∈ Ω P .

In this section, we sketch the idea leading to negative results on homogenization in the case of matched impedance. For precise statements and rigorous proofs, we refer to Theorem 5.1 in Section 5. In addition, we provide in Theorem 5.1 a positive homogenization results which is significantly stronger than Part (1) of Theorem 2.6.

A stochastic medium with matched impedance. We consider coefficients a = a ( x ) that are piecewise constant in the intervals, a ( x ) = a j for every x ∈ [ j , j + 1 ) =: I j . The numbers a j ∈ [ Λ − 1 , Λ ] are chosen as i.i.d. random variables. A possible choice is to pick a j according to a uniform distribution on the interval [ 1 , 2 ] . We set ρ j = 1 / a j for every j and ρ ( x ) := ρ j for all x ∈ I j . This construction guarantees a constant impedance, ρ ⋅ a = 1 on R .

Solutions of the wave equation. In this setting, there is an explicit formula for solutions. Let g : R → R of class C 2 be an arbitrary function with compact support in R + . We define a function u as follows: For every x > 0 and t ⩾ 0 , we set

u ( x , t ) := g ( x − j a j − t + ∑ i = 0 j − 1 1 a i ) for x ∈ I j ,

(2.11)

Before we verify the claim, let us calculate the initial values of u. For x < 0 , there holds u ( x , 0 ) = 0 . For 0 ⩽ x ∈ I j , there holds x = j + h for some h ∈ [ 0 , 1 ) and u ( x , 0 ) = g ( h / a j + ∑ i = 0 j − 1 ( 1 / a i ) ) . In particular, g does not coincide with the initial values of u.

Let us now show that, independent of initial data, u solves the wave equation. In the interior of the interval I j , we can calculate classically

ρ ∂ t 2 u ( x , t ) − a ∂ x 2 u ( x , t ) = ( ρ j − a j / a j 2 ⏟ = 1 / a j = ρ j ) g ″ ( x − j a j − t + ∑ i = 0 j − 1 1 a i ) = 0.

lim x ↗ j u ( x , t ) = g ( j − ( j − 1 ) a j − 1 − t + ∑ i = 0 j − 2 1 a i ) = g ( 1 a j − 1 − t + ∑ i = 0 j − 2 1 a i ) = lim x ↘ j u ( x , t ) .

lim x ↗ j a ( x ) ∂ x u ( x , t ) = g ′ ( 1 a j − 1 − t + ∑ i = 0 j − 2 1 a i ) = lim x ↘ j a ( x ) ∂ x u ( x , t ) ,

Homogenization. Let us now consider the rescaled coefficients a ε = a ( ⋅ / ε ) and ρ ε = ρ ( ⋅ / ε ) . A solution u ε of the wave equation is given by formula (2.11), which is modified to

u ε ( x , t ) = g ( x − ε j a j − t + ∑ i = 0 j − 1 ε a i ) for x ∈ [ ε j , ε ( j + 1 ) ) .

(2.12)

The effective parameter a ¯ is the harmonic average of a j , with our choices it is given by a simple expectation, a ¯ − 1 = ⟨ 1 / a j ⟩ for any j ∈ Z . The effective parameter ρ ¯ is the arithmetic average of ρ j , in our setting ρ ¯ = ⟨ ρ j ⟩ = ⟨ 1 / a j ⟩ = a ¯ − 1 . We see that the impedance of the effective medium is again 1 and that the effective speed is c ¯ = a ¯ . The effective wave equation is

1 a ¯ ∂ t 2 u ¯ − a ¯ ∂ x 2 u ¯ = 0.

(2.13)

u ¯ ( x , t ) = g ( x a ¯ − t ) .

(2.14)

We claim that stochastic homogenization fails in this setting on time intervals [ 0 , T 0 ε − β ] when we choose β > 1 . To see this, it suffices to compare u ε of (2.12) and u ¯ of (2.14). This is sufficient since, for t in bounded time intervals, there holds u ε ≈ u ¯ for small ε > 0 (and recall that rigorous results are presented in Section 5).

Calculating the difference. Let us assume that the support of g is contained in [ 0 , 2 ] , that the maximal value of g is 1 and that this maximum is attained in the point x = 1 . The solution u ¯ is a shift of the initial values, in this sense it is a wave that travels with speed a ¯ to the right. For every observation point x > 1 , the peak of the wave arrives at the time T ¯ ( x ) at which the argument of g in (2.14) is 1, i.e., T ¯ ( x ) := − 1 + ( x / a ¯ ) . In more mathematical terms: For x > 1 , the function u ¯ ( x , . ) : [ 0 , ∞ ) → R has its maximum at t = T ¯ ( x ) , the value in this maximum is 1.

We want to calculate the time instance T ¯ ε ( x ) at which the function u ε ( x , . ) : [ 0 , ∞ ) → R is maximal. The maximum of u ε ( x , . ) is at the point T ¯ ε ( x ) for which the argument of g in (2.12) is 1, hence

T ¯ ε ( x ) = x − j ε a j − 1 + ∑ i = 0 j − 1 ε a i ,

(2.15)

We are interested in the mismatch of the arrival times for the ε -solution and the homogenized solution,

Δ ε ( x ) := T ¯ ε ( x ) − T ¯ ( x ) = x − ε ⌊ x / ε ⌋ a j + ∑ i = 0 j − 1 ε a i − x a ¯ .

(2.16)

⟨ Δ ε ( x ) ⟩ = ⟨ − x a ¯ + ∑ i = 0 j − 1 ε a i ⟩ = ε j ( 1 a ¯ − ⟨ 1 a i ⟩ ) = 0.

(2.17)

For our analysis, it is not sufficient to calculate the averaged arrival time. Homogenization fails when, with a positive probability, we observe a wrong arrival time in the stochastic medium. Let us therefore calculate the typical size of the random variable Δ ε ( x ) . For the calculation we use the quantity σ ⩾ 0 , defined by the expectation σ 2 := ⟨ | ( 1 / a i ) − ( 1 / a ¯ ) | 2 ⟩ for any i. The number σ > 0 is the variance of the single entry in the sum. The independence of the random variables a i allows to calculate, again for x = ε j ,

⟨ | Δ ε ( x ) | 2 ⟩ = ⟨ | ε j a ¯ − ∑ i = 0 j − 1 ε a i | 2 ⟩ = ε 2 ⟨ | ∑ i = 0 j − 1 1 a ¯ − 1 a i | 2 ⟩ = ε 2 ⟨ ∑ i = 0 j − 1 | 1 a ¯ − 1 a i | 2 ⟩ = ε 2 j σ 2 .

(2.18)

| T ¯ ε ( x ε ) − T ¯ ( x ε ) | = | Δ ε ( x ε ) | = O ( ε 2 j ε σ 2 ) ⩾ O ( σ ε 1 − β ) .

(2.19)

In Theorem 2.6, we provide two critical parameters, β − and β + . The lower critical parameter, β − = β − ( γ ) = ( 1 − γ ) / ( 1 + γ ) , is related to positive homogenization results. It is derived in Section 3. The techniques of that section are well-established and independent of the dimension. It is possible to define a growth rate of correctors also in higher dimensions and to derive, with similar methods as in Section 3, positive homogenization results until time-scales ε − β − , where β − depends on the growth rate. We mentioned that recent advances in quantitative stochastic homogenization yield optimal estimates on the growth of the correctors for large classes of random media, see e.g. [5,15] and the references therein.

The upper critical parameter β + = β + ( γ ) = ( 1 − γ ) / γ is based on the construction of counter-examples. The counter-examples use either media with matched impedance or media-adapted domain transformations. In any case: If one considers layered media in dimension d > 1 , then the one-dimensional counter-examples still provide examples where homogenization does not hold. We mention that a restriction to initial values with compact support would still require some work, but we would not expect severe difficulties to the construction of counter-examples.

On the other hand, for stochastic media that exploit the full liberty of media in higher dimension, we do not have any counter-examples and they cannot be constructed easily following the ideas that are used in this contribution.